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Course by: Mark A. Davenport. E-mail the author

# Fourier Transforms as Unitary Operators

Module by: Mark A. Davenport. E-mail the author

## Fourier transforms as unitary operators

We have just seen that the DTFT can be viewed as a unitary operator between 2(Z)2(Z) and L2[-π,π]L2[-π,π]. One can repeat this process for each Fourier transform pair. In fact due to the symmetry between the DTFT and the CTFS, we have already established this for CTFS, i.e.,

CTFS: L 2 [ - π , π ] 2 ( Z ) CTFS: L 2 [ - π , π ] 2 ( Z )
(1)

is a unitary operator. Similarly, we have

CTFS: L 2 ( R ) L 2 ( R ) CTFS: L 2 ( R ) L 2 ( R )
(2)

is a unitary operator as well. The proof of this fact closely mirrors the proof for the DTFT. Finally, we also have

DFT: C N C N . DFT: C N C N .
(3)

This operator is also unitary, which can be easily verified by showing that the DFT matrix is actually a unitary matrix: UHU=UUH=IUHU=UUH=I.

Note that this discussion only applies to finite-energy (2/L22/L2) signals. Whenever we talk about infinite-energy functions (things like the unit step, delta functions, the all-constant signal) having a Fourier transform, we need to be very careful about whether we are talking about a truly convergent Fourier representation or whether we are merely using an engineering “trick” or convention.

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